Monotone and Pseudo-Monotone Equilibrium Problems in Hadamard Spaces

TL;DR

This paper investigates equilibrium problems in Hadamard spaces, establishing existence, approximation methods, and convergence results for solutions related to pseudo-monotone bifunctions, with applications in fixed point theory and convex minimization.

Contribution

It introduces new existence and convergence results for equilibrium problems in Hadamard spaces, including proximal point algorithms and regularization techniques, extending nonlinear analysis tools.

Findings

Proved existence of solutions under pseudo-monotonicity in Hadamard spaces.

Established convergence of proximal point algorithms to equilibrium points.

Demonstrated strong convergence without extra assumptions using Halpern-type regularization.

Abstract

We study equilibrium problems in Hadamard spaces, which extend variational inequalities and many other problems in nonlinear analysis. In this paper, first we study the existence of solutions of equilibrium problems associated with pseudo-monotone bifunctions with suitable conditions on the bifunctions in Hadamard spaces. Then to approximate of an equilibrium point, we consider the proximal point algorithm for pseudo-monotone bifunctions. We prove existence of the sequence generated by the algorithm in several cases in Hadamard spaces. Next, we introduce the resolvent of a bifunction in Hadamard spaces. We prove convergence of the resolvent to an equilibrium point. We also prove $\bigtriangleup$-convergence of the sequence generated by the proximal point algorithm to an equilibrium point of the pseudo-monotone bifunction and also the strong convergence with additional assumptions on the bifunction. Finally, we study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point without any additional assumption on the pseudo-monotone bifunction. Some examples in fixed point theory and convex minimization are also presented.